J . Chem. SOC., Faraday Trans. I , 1982, 78, 3529-3535 Theory of Volumetric Behaviour of Hydrous Melts The Systems LiN0,-H,O and NH,N03-H,O BY ZDENEK KODEJS Institute of Inorganic Chemistry, Czechoslovak Academy of Sciences, Majakovskeho 24, 160 00 Prague 6, Czechoslovakia AND GIUSEPPE A. SACCHETTO* Institute of Analytical Chemistry of the University, Via Marzolo 1, 35100 Padova, Italy Received 1st March, 1982 An equation for the dependence of the molar volume of a hydrous melt on its molar composition is derived on the basis of the concept of the apparent molar volume of water and from a previous calculation of the distribution of water molecules between sites near ions and sites near water molecules in the melt. The densities of LiN0,-H,O and NH,NO,-H,O liquid mixtures have been measured at 333 K over the composition ranges 0.05 < x(LiN0,) c 0.3 and 0.1 < x(NH,NO,) < 0.5, and have been used, together with literature data for the systems NH,NO,-H,O, (Lin,5 K,,,)NO,-H,O, Ca(NO,),-H,O and Cd(NO,),- H,O, to test the validity of the derived equation and to calculate the values of the physical parameters involved.Many investigations of the volumetric behaviour of electrolyte solutions have confirmed the suitability of the concept of the apparent molar volume (a.m.v.) of a sa1t.l All deviations from the additivity of V z (the molar volume of the pure solvent) and V: (the molar volume of the pure salt) are included in the value of the a.m.v. of the salt, which then depends on the salt concentration. At the other end of the composition range, i.e.for the so-called ‘hydrous melts’, and for their non-aqueous analogue^,^ the concept of the a.m.v. of the salt cannot be applied. For this type of system a different approach has been proposed by Kodeji and Slams,* who formulated the alternative concept of the ‘apparent molar volume of water’, based on the assumption that the following equation holds for the total volume of the solution: V = n, Vw + n, V,O where Vw is the a.m.v. of water. In highly concentrated solutions the fraction of water molecules that have the typical bulk water structure is rather small, as most of the water is strongly influenced by the ions. The concept of the a.m.v. of water has been used to derive a relationship between the molar volume of the melt, V,, and its composition [see eqn (2) of ref.(4)], which for a binary salt-water system at fixed temperature reduces to the following expression : where x, is the mole fraction of the salt, R , is the mole ratio of water (i.e. R , = nw/ns), and A , C and D are empirical parameters. The main scope of the present paper is to provide a theoretical explanation of the physical meaning of the parameters in eqn (2) and to show how the general features 35293530 VOLUMETRIC BEHAVIOUR OF HYDROUS MELTS of the dependence of V, on x, can be predicted on this basis. The proposed treatment starts from a model for solutions of water in molten-salt mixtures recently developed by Sacchetto and KodejL5 Density measurements for the LiN0,-H,O and NH,NO,- H,O systems have also been performed and the density data are used to verify the ability of the derived equation to fit experimental molar-volume data with its parameters taking physically reasonable values.Further tests on the following systems are performed using literature data : NH,N0,-H,0,6 (Lio.5 K0,5)N03-H20,7 Ca(NO,),- H,Os and Cd(NO,),-H,O.g THEORETICAL The quantity within square brackets in eqn (2) represents the concentration- dependent molar volume of water, V,, in a salt-water liquid mixture: By introducing expressions for A and C which are derived at the concentration limits of the binary salt-water system, and by replacing R , with (1 -x,)/x,, the following equation is obtained for V, : 1 -x, 1 + x,(D - 1) vw" vw = Dxs v,"+ 1 +x,(D- 1) (4) where V$ is the molar volume of water at infinite dilution in the liquid salt.Eqn (4) is formally very similar to an equation recently derived by Sacchetto and KodejS5 by means of a statistical-mechanical calculation based on the quasi-lattice model used to explain the composition dependence of enthalpies of transfer of water from liquid water to infinitely dilute solutions of water in molten-salt mixtures. They assumed two types of sites to be available for a water molecule entering the liquid salt mixture AY-BY, and calculated the equilibrium distribution of the water molecules between these two types; the following equation for the enthalpy of transfer of water from liquid water to a binary molten-salt mixture, AHgix, was obtained: where the quantities Z are quasi-lattice coordination numbers, and AH,, are enthalpies of transfer of water from liquid water to the pure molten salts.The quantity/? is defined as and 0 = ZAy/ZBy. The statistical weighting factors for AH&y/ZAy and AH,B,y/Z,, in eqn ( 5 ) are formally similar to those for V$ and V z in eqn (4). There remains only to correlate the product PO with the parameter D. We can suppose that in a liquid salt-water mixture two types of sites having different energy levels are available to accommodate water molecules. One type of site is in the immediate neighbourhood of the ions of the salt; the effective molar volume of water at this type of site may be regarded as the molar volume of water at infinite dilution in the molten salt, V,". The second type of site is in the bulk water structure; the effective molar volume of water at this site is the same as the molar volume of pure water, V:.The different effective molar volumes of water at the two types of site may essentially be ascribed to the differentz. KODEJS AND G. A. SACCHETTO 353 1 modes and densities of packing of the water molecules in the hydration spheres of the ions and in bulk water. The distribution of water molecules between these two types of site is governed by statistical factors which are essentially the same as those calculated in our previous paper5 [see eqn (5)]. Thus the effective molar volume of water in a salt-water mixture, V,, results from a non-linear combination of V z and V;, as given by eqn (4). The parameter D is given by PO; in the present case AHSa't = exp( -2) Zsalt RT (7) as the enthalpy of transfer of water from liquid water to bulk water is equal to zero [cf.eqn (6)], and 0 is defined as the ratio Zsalt/Zwater; the quantity Zwater can be interpreted as an average coordination number of water in liquid water and an approximate value of 4 can be taken from models of liquid-water structure.1° Values of 2 of ca. 4-5 were found to be acceptable for various nitrates in ref. (5). The value of 0 is then taken approximately equal to 1. By combining the assumption r/s = V,O [see eqn (l)] with eqn (4), where D is replaced by PO, we have the following equation for the dependence of the molar volume of a liquid salt-water mixture on the mole fraction of the salt: 1 -x, 1 +x,(ae- 1) v:) (1 - x s ) v, = v,ox,+ ( vz+ 1 +x,(a@- 1) where P is given by eqn (7) and 0 is assumed equal to 1 (see above). EXPERIMENTAL Reagent-grade LiNO, and NH,NO, (Merck) were vacuum desiccated and stored over P,O,.The densities of the aqueous solutions were determined by means of a DMA digital densimeter (A. Paar K.G., Austria) ; the experimental procedure and precision of the measurements have been discussed previ~usly.~ The densities of the LiN0,-H,O and NH,NO,-H,O solutions were measured at 333 K in concentration intervals ranging from xs = 0.05 and 0.1, respectively, up to the saturation limits. The experimental density data are summarized in table 1, together with the corresponding molar volumes of the mixtures. TABLE EXPERIMENTAL VALUES OF DENSITY AND MOLAR VOLUME OF THE LiN0,-H,O AND NH4N03-H20 SYSTEMS AT 333 K LiN0,-H,O NH,NO,-H,O d vm d vm X S /g cm-, /cm3 mol-l XS /g cm-, /cm3 mo1-I 0.05 1.0819 19.00 0.10 0.10 1.1721 19.72 0.15 0.15 1.2547 20.45 0.20 0.20 1.3303 21.20 0.25 0.25 1.3990 21.98 0.30 0.30 1.4613 22.78 0.35 0.40 - 0.45 0.50 - - - - - - - - 1.1177 1.1695 1.2129 1.2497 1.2825 1.3 105 1.3383 1.3627 1.3870 21.67 23.36 25.08 26.83 28.56 30.31 32.00 33.71 35.353532 VOLUMETRIC BEHAVIOUR OF HYDROUS MELTS I 0 0.5 1 .o x, FIG.1 .-Graphical simulation of the dependence of the molar volume of salt-water mixtures on the mole fraction of salt, as derived from eqn (8) (molar volume is expressed in cmS mol-l, enthalpy in kJ mol - I ) . V," = 25, AHtr = 0; (d), Vg = 15, AHtr = - 16; (e), V," = 15, AHtr = 16. T = 333 K, 2 = 4, B = 1 , V: = 40, V," = 20. (a), VF = 15, AHtr = 0; (b), VF = 20, AHtr = 0; (c), RESULTS AND DISCUSSICN SIMULATION It is of interest to show graphically how the main features of the dependence of Vm on x,, as given by eqn (8), vary with the parameters.Let us first discuss the simple case where AH,, = 0. There are three different possibilities according to the relative magnitudes of V: and V z : if V z = V:, we get a linear function for Vm as a function of x, [i.e. an additivity relationship between V: and V,O, see line (b) in fig. I] ; if V z < V,", negative deviations from simple add'tivity are obtained [line ( a ) ] ; if V z > V,", positive deviations are obtained [line (c)]. Only few water transfer-enthalpy data are available (see below). The values of AH,, found are usually negative, but positive values can be also found.On the other hand, it is reasonable to consider that the molar volumes of water at the two above-mentioned types of site are generally different. The most frequent case for real systems is certainly Let us now suppose that V: < V:. A negative value for AH,, not only enhances the negative deviations from additivity [cJ curve ( d ) with curve (a)] but also introduces a significant asymmetry effect, so that the deviations are more pronounced in the low-x, region than in the high-x, region. In the opposite case, when a positive value for AH,, is chosen the negative deviations from additivity are reduced [cf. curve (e) with curve (a)] and an asymmetry effect is also evident in this case. A positive value of AHtr is known only for the NH,NO,-H,O system (see below); only very small devia- tions from additivity are expected in this case.For the less frequent case where V$ > V: the deviations from additivity are expected to be positive in any case. Different asymmetric shifts may also be found in this case, depending on the sign of AHt,. We can assert that the only condition for perfect additivity of the molar volumes V$ < V:.'oz. K O D E J S A N D G. A. SACCHETTO 3533 of water and salt is V$ = V;, whatever the value of AHtr. However, a good apparent additivity can be obtained, in spite of the inequality of V," and V:, if Alltr has a positive and sufficiently large value. This situation is rather infrequent in practice (see below for the NH,NO,-H,O system). We can thus say that the additivity of molar volumes of liquid salt-water mixtures should be the exception rather than the rule.COMPARISON WITH EXPERIMENTAL AND LITERATURE DATA To test the applicability of eqn (8), we first use the experimental density data for LiN0,-H,O and for NH ,NO,-H,O solutions as obtained from our measurements (table 1). Further density-data sets were collected from the literature for the systems NH,N0,--H,0,6 (Li, K, 5)N0,-H,0,5 Ca(NO,),-H,OH and Cd(NO,),-H,Og (see table 2). We first decide which of the parameters of eqn (8) can be considered as known a priori and which can be evaluated from the fit. Some of them are in fact known with a good degree of accuracy: e.g. the temperature and the molar volume of liquid water.'l The value of 2 is taken as 4 (see above); however, it does not play a decisive role as its choice influences only the term AHt,./Z, whose uncertainty comes mainly from the value of AH,, (see below).Difficulties arise in the evaluation of AHtr for several systems. In fact, only in the case of LiN0,-H,O and NH,NO,-H,O have transfer-enthalpy values been correctly evaluated1, by applying the Stokes-Robinson ' adsorption-hydration' mode113 to vapour-pressure data and by taking into account the 'internal' entrcpy contribution to the water-transfer process., For the other systems reported in table 2 approximate A l l t r values can be estimated from E, - El data (the 'net' adsorption energy13), which are available in the literature. The contribution of the 'internal' entropy change is estimated to be ca. 40-500/, of the total E,- El effect in the case of the LiN0,-H,O system.l2?l4 On applying the same correction to the Ea-E, values for the (Lio.5 KO S)NO,-H,O and the Ca(NO,),-H,O systems1, we obtained the AHtr values given in table 2.column 3. For the Cd(NO,),-H,O system, for which directly measured E.', - E, values are not available, we calculated an approximate AHtr value from that for the Ca(NO,),-H,O system by using the quotient 0.9, which is obtained by dividing the extrapolated value of Ea-El for the calcium-containing system by that of the cadmium-containing system as reported by Sangster et ~ 1 . ~ ~ 7 16* The remaining parameters V$ and V': were instead obtained from the fitting treatment and are reported in table 2, columns 4 and 5, respectively. For the Ca(NO,),-H,O and Cd(NO,),-H,O systems the best fit of eqn (8) and more reliable values of the parameters V," and V,P were obtained by using equivalent volumes and equivalent fractions instead of the corresponding molar quantities.This approach has been adopted previously by other authors in treating density data for ultraconcentrated solutions of bi-univalent electrolytes [see e.g. ref. (1 7)]. We have tested the effect of the uncertainties inherent in the transfer-enthalpy values on the calculated values of the parameters V'$ and V:. We found that changing the AH,,/Z values by 30p/,, a change large enough to cover the most unfavourable case, results in small changes in V:, not exceeding 2:{; the changes in V'Z are larger, but they never exceed 10%. The V: values resulting from the fitting treatment can be compared with the molar * The E,- E, values for the Ca(NO,),-H,O and Cd(NO,),-H,O systems given by these authors are the result of extrapolations over very wide concentration ranges pzrformed on data for aqueous solutions of mixtures of (Ago ,Tl,,,)NO, with Ca(NO,), and with Cd(NO,),, respectively; for this reason they are not reliable.Moreover, the datum for the calcium-containing system does not agree with that given in ref. (14). The quotient 0.9 adopted above is perhaps more reliable.3534 VOLUMETRIC BEHAVIOUR OF HYDROUS MELTS TABLE 2.-vALUES OF THE PARAMETERS OF EQN (8) AT 333 K salt A& v: V; Vs"d x, range /kJ mol-l /cm3 mol-1 /cm3 mol-l /cm3 mol-1 NH,NO, 0.1-0.5 4.0 17.1 53.0 - LiNO,-KNO,b 0.5-0.8 - 4.2 15.4 43.5 43.6e 0.05-0.27 - 8.4 13.3 37.lC 36.3f NH,NO,a 0.25-0.45 4.0 15.9 53.4 - LiNO, 0.05-0.3 - 7.3 16.5 35.1 33.4e CdfN03)2 0.05-0.35 - 7.5 12.4 37.4c - V: is 18.32 cm3 mol-l and 19.09 cm3 m o t 1 at 333 K and 392 K, respectively;13 a data from ref. (6); (L& K,,,)N03 at 392 K; equivalent volume; values extrapolated from density data reported in the indicated references; ref, (18); f ref. (19). 0 0.2 0.4 0.6 0.8 1.0 xs FIG. 2.-Comparison between experimental and calculated deviations of molar volumes of mixtures from purely additive behaviour. 0, NH,N03-H,O (this work); a, NH,NO,-H,O [ref (6)]; 0, LiN0,-H,O; +, (Li,,,K,.,)N03-H,0; A, Ca(NO,),-H,O; x , Cd(NO,),-H,O. volumes of the undercooled liquid salts calculated from their densities as extrapolated on the basis of density-temperature equations for the molten salts when available in the literature (table 2, column 6).No such data are available for NH,NO, and Cd(NO,),. The agreement between the two sets of V,O values may be regarded as satisfactory, in view of the large uncertainty which affects the above procedure. The larger discrepancy between the V . values for LiNO, can be ascribed to the rather narrow concentration range over which the density could be measured for the LiN0,-H,O solutions. The equivalent volume obtained for Cd(NO,), is almost equal to that for Ca(NO,),. Since the two cations are similar in size, this finding provides a reasonable justification for the calculation presented. From table 2 the V$ values are seen to decrease as the ionic potential of the cation increases.The (L& Ko.,)NO,-H,O system does not fit into the observed pattern; however, the value of V," is rather uncertain, as the measurements were restricted toz. KODEJS AND G. A. SACCHETTO 3535 the salt-rich concentration range. The observed pattern also provides support for the value of V z for the NH,NO,-H,O system calculated from our data, in contrast to that calculated from the data of Sharma and Gaur? The scatter of the experimental V, data about the calculated V, against x, curves can be seen in fig. 2, where deviations, AVm, from the linear additive behaviour of V: and V,O are shown by way of illustration. The fit is good for all systems except NH,NO,-H,O, for which systematic trends are apparent in both sets of data. The small, but not negligible, systematic discrepancy between our data and those of Sharma and Gaur6 is reflected in the above-mentioned discrepancy in Vg values.This work was carried out within the framework of an agreement between the National Research Council of Italy (C.N.R.) and the Czechoslovak Academy of Sciences (c. S .A .V.). F. J. Millero, in Water and Aqueous Solutions, ed. R. A. Horne (Wiley-Interscience, New York, 1972), chap. 13. * J. Braunstein, in Ionic Interactions, ed. S. Petrucci (Academic Press, New York, 1971), vol. I, chap. IV. G. A. Sacchetto, G. G. Bombi and C. Macca, J . Electroanal. Chem. Interfacial Electrochem., 1974, 50,300; Z. KodejS, G. A. Sacchetto, C. Macca and G. G. Bombi, Ann. Chim. (Rome), 1978,68, 151. Z . KodejS and I. Slama, Collect. Czech. Chem. Commun., 1980, 45, 17. G. A. Sacchetto and Z . KodejS, J . Chem. Soc., Faraday Trans. I , 1982, 78, 3519. R. C. Sharma and H. C. Gaur, J , Chem. Eng. Data, 1977, 22, 41. W. W. Ewing and R. J. Mikovsky. 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